Average Error: 43.7 → 0.7
Time: 10.7s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y
double f(double x, double y) {
        double r39836 = x;
        double r39837 = exp(r39836);
        double r39838 = -r39836;
        double r39839 = exp(r39838);
        double r39840 = r39837 + r39839;
        double r39841 = 2.0;
        double r39842 = r39840 / r39841;
        double r39843 = y;
        double r39844 = cos(r39843);
        double r39845 = r39842 * r39844;
        double r39846 = r39837 - r39839;
        double r39847 = r39846 / r39841;
        double r39848 = sin(r39843);
        double r39849 = r39847 * r39848;
        double r39850 = /* ERROR: no complex support in C */;
        double r39851 = /* ERROR: no complex support in C */;
        return r39851;
}
↓
double f(double x, double y) {
        double r39852 = 0.3333333333333333;
        double r39853 = x;
        double r39854 = 3.0;
        double r39855 = pow(r39853, r39854);
        double r39856 = 0.016666666666666666;
        double r39857 = 5.0;
        double r39858 = pow(r39853, r39857);
        double r39859 = 2.0;
        double r39860 = r39859 * r39853;
        double r39861 = fma(r39856, r39858, r39860);
        double r39862 = fma(r39852, r39855, r39861);
        double r39863 = 2.0;
        double r39864 = r39862 / r39863;
        double r39865 = y;
        double r39866 = sin(r39865);
        double r39867 = r39864 * r39866;
        return r39867;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.7

    \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  2. Simplified43.7

    \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]

  3. Taylor expanded around 0 0.7

    \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2} \cdot \sin y\]

  4. Simplified0.7

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}}{2} \cdot \sin y\]

  5. Final simplification0.7

    \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))